Download - Bulk viscosity of interacting Hadrons
BULK VISCOSITY OF INTERACTING HADRONS
Anton Wiranata and Madappa Prakash
Department of Physics and AstronomyOhio University, Athens, OH
Quark Matter 2009, March 30 – April 4, Knoxville, TN
Quark Matter 2009, March 30 – April 4, Knoxville, TN
Topics Overview of results Bulk viscosity in the Chapman–Enskog
approximation Bulk viscosity and speed of sound
Analysis in limiting situations Numerical results of ηv and ηv /s
Chiral pions Massive interacting pions
Multi-component mixturesBinary and N-species systems
Ongoing work and future plans
Bulk viscosity / entropy density vs T
Kapusta, arxiv:0809.3746v2
Quark Matter 2009, March 30 – April 4, Knoxville, TN
Wiranata & Prakash.
Chen and Wang, Arxiv:0711.4824v1
P.Arnold, et.al JHEP05, 051(2003) Kharzeev & Tuchin
JHEP09, 093(2008)
Chapman-Enskog Approximation “For small deviations from equilibrium, the distribution
function can be expressed in terms of hydrodynamic variables ( f(x,p) μ(x), u(x), T(x) ) and their gradients. Transport coefficients (viscosities, thermal conductivity, etc. ) are then calculable from relativistic kinetic theory.”
Equilibrium distribution function
Quark Matter 2009, March 30 – April 4, Knoxville, TN
μ(x) : Chemical potentialu(x) : Flow velocityT(x) : Temperature
φ(1) : Deviation function
Beginning with the relativistic Boltzman equation
and inserting f(0)
the solution (deviation function) has the general structure
Quark Matter 2009, March 30 – April 4, Knoxville, TN
Bulk ViscosityFirst approximation :
Reduced enthalpy : (ε + P) / nmc2
Relativistic Omega Integrals
Transport cross sectionThermal weightRelative momentum weight
Relative momentum : g = mc sinh ψTotal momentum : P = 2mc cosh ψ
Quark Matter 2009, March 30 – April 4, Knoxville, TN
Ratio of specific heats
Bulk Viscosity & the Speed of Sound
Chapman-Enskog 1st approximation
Quark Matter 2009, March 30 – April 4, Knoxville, TN
Utilizing
Features thermodynamic variables
The omega integral contains transport cross-section
Analysis for limiting cases
Nearly massless particles or very high temperatures (z << 1) :
Massive particles such that z >> 1 :
Lesson : For a given T, intermediate mass particles contribute significantly to ηv
Quark Matter 2009, March 30 – April 4, Knoxville, TN
Variational approximation Sean Gavin, Nucl. Phy. A 435, 826 (1985)
Gavin’s result can be rewritten as
Kapusta , arxiv:0809.3746v2
A similar analysis leads to the result that intermediate mass particles controlthe magnitude of ηv Quark Matter 2009, March 30 – April 4, Knoxville, TN
Numerical results (interacting pions)
Parametrization from Bertsch et al. , PR D37 (1988) 1202.
Quark Matter 2009, March 30 – April 4, Knoxville, TN
Entropy Density
15
Non-interacting part
Interacting part (up to 2nd virial expansion)
Entropy Density
Energy Density
Pressure
Fugacity
β =1/kT : Inverse Temperature
InteractionsThermal weightKinematic Factor
Quark Matter 2009, March 30 – April 4, Knoxville, TN
``Venugopalan & Prakash Nucl. Phys. A546 (1992) 718''
Results (Interacting Pions)
16
ηv 3rd appro x 103
2nd
1st
ηs 1st appro
2nd
T[MeV]
[GeV
][fm
-2][c
-1]
Total
Non-in
terac
ting
Interacting
T[MeV]
[fm-3]
Entropy DensityBulk &Shear Viscosity
Prakash et al. ,Phys. Rep. 227 (1993) 321
Quark Matter 2009, March 30 – April 4, Knoxville, TN
Chen and Wang, Arxiv:0711.4824v1
Bulk viscosity / entropy density
Quark Matter 2009, March 30 – April 4, Knoxville, TN
QGP sum rule
LatticeMassless pions
For chiral pions ηv/s increases with T in contrast to ηv /s for massive interacting pions
Wiranata & Prakash
Multi-Component system
Transport cross sectionThermal weightRelative momentum weight
W.A. Leeuwen, et.al
Begin with binary mixture
Ratio of specific heats
Quark Matter 2009, March 30 – April 4, Knoxville, TN
Binary mixtureResults for constant cross-sections
Quark Matter 2009, March 30 – April 4, Knoxville, TN
Multi-component systemResult for N – species at pth order of approximation
Solubility conditions (assures 4-momentum conservation in collisions)
Coefficients to be determined Omega integral
Involves ratios of specific heats
Quark Matter 2009, March 30 – April 4, Knoxville, TN
1st order calculation
Coefficients to be determined
Quark Matter 2009, March 30 – April 4, Knoxville, TN
Ongoing work and future plans Calculation of ηv & ηv /s for a mixture of interacting
hadrons with masses up to 2 GeV. Development of an approach to calculate the needed
differentialcross-sections for hadron-hadron interactions including resonances up to 2 GeV.
Comparison of the Chapman-Enskog results with those of the Green-Kubo approach.
Inclusion of decay processes. All the above for shear viscosity ηv and ηv /s
Quark Matter 2009, March 30 – April 4, Knoxville, TN
Thank you
Quark Matter 2009, March 30 – April 4, Knoxville, TN